Approximate Homomorphisms of Ternary Semigroups

TL;DR

This paper investigates the stability of approximate ternary homomorphisms between commutative semigroups and Banach spaces, establishing conditions under which these mappings are stable or superstable.

Contribution

It extends the Hyers--Ulam--Rassias stability theory to ternary semigroup homomorphisms into Banach spaces and proves superstability in Banach algebras with multiplicative norms.

Findings

Proves generalized Hyers--Ulam--Rassias stability for commutative semigroup mappings.

Establishes superstability of ternary homomorphisms into Banach algebras.

Provides conditions for stability and superstability of approximate homomorphisms.

Abstract

A mapping $f:(G_1,[ ]_1)\to (G_2,[ ]_2)$ between ternary semigroups will be called a ternary homomorphism if $f([xyz]_1)=[f(x)f(y)f(z)]_2$. In this paper, we prove the generalized Hyers--Ulam--Rassias stability of mappings of commutative semigroups into Banach spaces. In addition, we establish the superstability of ternary homomorphisms into Banach algebras endowed with multiplicative norms.